keith a mclauchlan-molecular physical chemistry_ a concise introduction-royal society of chemistry...

8/17/2019 Keith a McLauchlan-Molecular Physical Chemistry_ a Concise Introduction-Royal Society of Chemistry (2004)

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Molecular Physical Chemistry

A Concise Introduction


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Molecular Physical hemistry

A

oncise

Introduction

K A

McLauchlan

University

o

Oxford

U K

dv ncing

the chemic l

sciences


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ISBN 0-85404-619-4

catalogue record for this book is available from the British Library

he Royal Society of Chemistry 2004

ll rights reserved

Apart fr om air dealing-for the purposes

o

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Act

1988

and the Copy right and Related Righ ts Regulations 2003 this puhlicution m ay

not be reproduced stored or transm itted in an y or m or by any means without the prior

permission in writing of Th e Ro yal Society of Chemistry or in the case of reproduction in

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or in accordance with the terms o f t h e licences issued by the appropriate Rep rodu ction

Rights Organization outside the U K . Enquiries concerning reproduction outside th e terms

stated here should be sent t o T he Royal Socie ty of Chem istry at the address printed on this

Page.

Published by The Royal Society

of

Chemistry

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reface

To

write any b ook is an indulgence and my excuse for having do ne

so

is

tha t have tried to provide underg raduates with a text that differs in

approa ch from any other I kn ow o n its subject ma tter. It is an attempt to

provide the reader with a n understanding of thermodynamics and to a

lesser extent) reactions based upo n ato m s and molecules an d their pr op-

erties rather th an o n one based up on the historical development of these

subjects. Th is is possible as a result

of

innovative experiments performed

on very small numb ers of ato m s an d molecules that have been performed

in the last decade o r so

This book makes no pretence to be a primary source of its subject

matter. Rather it attempts to give molecular insight into the familiar

equ ation s of therm odyna mics, for example, and should be read in con-

junctio n with the excellent Physical Chem istry texts th at a lready exist. It

is an aid to understanding, no more and n o less. But hope that those

deterred by the elegant but possibly dry approaches found in other

books, which develop the subject without the properties of molecules

considered, will find this m ore to their liking. Th e subjects are imp ortan t

to the whole understand ing

o

Physical Chem istry an d provide the unde r-

lying philosophical structure th at binds its apparently separate subjects

together.

I a m indebted to all my students

or

what they have taught me an d for

the sheer pleasure of kno w ing them . But my g reatest deb t is to Jo an , for

everything imp ort an t in my life.

V


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Contents

Chapter Some Basic Ideas and Exam ples

1 1 Introduction

1.2 Energies and Heat Capacities of Atom s

1.3 Heat Capacities of Diatomic M olecules

1.4 Spectroscopy an d Qu antisation

1.5 Summary

1.6 Fu rthe r Implications from Spectroscopy

1.7 Nature of Quantised Systems

1.7.1 Boltzmann Distribu tion

1.7.2 Two level Systems

1.7.3 Two level Systems with Degeneracies Gr eate r tha n

Unity; Halogen Atoms

1.7.4 Molecular Exam ple: NO gas

Appendix 1 1 The Eq uipartition Integral

Appendix 1.2 Term Symbols

Problems

Chapter

2

Partition Functions

2.1 Mo lecular Partition Fun ction

2.2 Boltzman n Distribution

2.3 Canonical Partit ion Fun ction

2.4 Sum mary of Partition Fun ctions

2.5 Evaluation of Molecular Partition Functions

2.5.1 Electronic Pa rtition Fu nc tion

2.5.2 Vibrational Partition Fu nction

2.5.3 Th e Rotational Partit ion Function

2.5.2.1

2.5.3.1

2.5.3.2

Vibrational Heat Capacity

of

a Diatomic Ga s

Ro tational Heat Capacity of a Diatomic G as

Ro tational Partition Functions in the Liquid

State

1

2

5

10

14

14

16

18

19

22

24

26

26

28

3

30

33

36

39

39

40

41

44

46

51

53

vii


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viii

Contents

2.5.4

Translational Partition Function

54

2.5.4.1 The Translational Heat Capacity

55

Appendix

2.1

Units

57

Problems 58

2.6

Overall Molecular Partition Function for a Diatomic Molecule 56

Chapter

3

Thermodynamics

3.1 Introduction

3.2 Entropy

3.3

3.4 State Functions

3.5

Thermodynamics

3.5.1 First Law

Entropy at 0 K and the Third Law

of

Thermodynamics

3.5.1.1

3.5.2 Second Law

Thermodynamic Functions and Partition Functions

Absolute Entropies and the Entropies of

Molecular Crystals at 0 K

3.6 Free Energy

3.7

3.8 Conclusion

Problems

Chapter Applications

4.1 General Strategy

4.2 Entropy

of

Gases

4.2.1

Entropies of Monatomic Gases

4.2.2 Entropy of Diatomic and Polyatomic Molecules

Two level Systems; Zeeman Effects and Magnetic Resonance

4.3.1 Internal Energy of Two level Systems

4.3.2

Curie Law

Pauli Principle and Ortho and Para hydrogen

4.6.1 Isotope Equilibria

4.3

4.4 Intensities of Spectral Lines

4.5

4.6 Chemical Equilibria

4.7 Chemical Reaction

4.8 Thermal Equilibrium and Temperature

Problems

Chapter

5

Reactions

5.1 Introduction

5.2 Molecular Collisions

5.2.1 Collision Diameters

5.2.2 Reactive Collisions

9

59

59

61

63

63

64

61

69

70

72

74

74

77

77

77

78

79

80

80

81

82

84

88

92

93

97

99

1 1

101

101

102

105


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Contents

ix

5.3

Collision Theory

5.4 Energy Considerations

5.4.1

Activation Energy

5.4.2

Disposal

of

Energy and Energy Distribution in

Molecules

5 5 Potential Energy Surfaces

5 6

Summary

Problems

Answers to Problems

Some Useful Constants and Relations

Further Reading

108

110

110

111

112

115

116

8

2

22

Subject Index

23


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CHAPTER

Some Basic Ideas and Examples

1.1

INTRODUCTION

Phy sical chem istry is widely perceived a s a collection of largely indepen-

dent topics, few of which appear straightforward. This book aims to

remove this misconception by basing it securely on the atom s a n d mol-

ecules that c onstitute matter, an d their properties. W e shall concen trate

o n just two aspects and we focus mainly o n thermodynam ics, which

although extremely powerful is one of the least popular subjects with

students. A briefer account describes how reactions occur. We shall

nevertheless encounter the major building blocks

of

physical chem istry,

the foun dation s that, if unde rstood , togethe r with their inter-dependence ,

remove any mystique. These include statistical thermodynamics, ther-

mody namics and qua ntum theory.

T he way t ha t physical chem istry is tau gh t tod ay reflects the historical

process by which und erstan ding w as initially obtain ed. On e subject led to

another, not necessarily with any underlying philosophical connection

but largely as a result of what was possible at the time. All experiments

involved very large numbers of molecules (although when ther-

modynamics was first formulated the existence of atoms and molecules

was not generally accepted) and people attempted to decipher what

hap pen ed a t a mo lecular level from their results. Th is was very indirect.

Nowadays the existence and properties of atoms and molecules are

established and experiments can even be performed o n individual atom s

and molecules. This provides the opportunity for a different way of

looking at the subject, building from these properties to deduce the

characteristic behaviour of large collections of them, which is more in

keeping with how chemistry is taught at school level. Similarly, our

understanding of how reactions occur has come from observations of

samples containing -hu ge num bers of m olecules a nd we have tried to

deduce w hat h app ens a t molecular level from them. Yet it is now possible

to o bserve reactions between individual p airs of molecules, an d we can

reverse the procedure and start from these observations to understand

1


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Chapter

1

reactions in bulk. It is the object of this boo k to d em ons trate th e possibil-

ity of a molecular app roac h t o thermody nam ics an d reaction dynam ics.

It is not intended as a n introduction to these subjects but rathe r is offered

as an aid to understand ing them, with some prior know ledge assumed.

W e start with the properties of ato m s and molecules as deduced from

thermod ynam ic measurem ents an d from spectroscopy. This is, paradoxi-

cally, the historical approach but it establishes straightaway that the

properties are directly connected t o the thermod ynam ics an d it is artifi-

cial to separate the two. B ut once the conne ction is established we show

how it ca n be exploited to give real insight into various problem s. In this

chapter we introduce the fact that the energy levels of atoms and mol-

ecules are quantised an d use som e simple ideas to establish the effective-

ness of our general approach before proceeding to their origins in the

second chapter.

1.2

ENERGIES A N D HEAT CAPACITIES O F ATOMS

In the gas phase, ato m s move freely in space a nd frequently collide, at a

rate tha t de pends up on the pressure of the gas. At atmo spheric pressure

l o 5

N

m-2) and room temperature they move approximately

100

mo lecular diam eters between collisions, a t average velocities abo ut eq ual

to that

of

a rifle bullet

(300

m

s-l).

In elastic collisions some atoms

effectively stop whilst others gain increased velocity cJ: collisions of

billiard balls) so that instead of all the a to m s having a single velocity they

have a wide distribution

of

velocities. This is the familiar Maxwell dis-

tribution (F igure

1.1)

ha t results from classical Ne wto nian mechanics. In

it all velocities are possible but som e are more probable than others. The

mo st probable velocity depends upon the temperature, as does the width

of the distribu tion.

A moving atom

of

mass m possesses a kinetic energy of +mu2,where u is

its velocity. Since in the whole collection of atoms in a gas there is no

restriction t o th e velocity of a n a to m , there is no restriction to its energy

either. Using the Maxwell distribu tion (see below), the average energy of

an ato m can be shown to be

where

2

s the mean square velocity of the atoms in the sample, k is

Boltzmann’s co nstan t

k

=

R/N,

where

R

is the universal gas constant

and NA the Avogadro number) and T is the absolute temperature. To

obta in the to tal energy, E , of a mole of gas we simply multiply by th e to tal

number

of

atoms,

NA,

an d obta in

( )RT.

This is the energy due to the


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Some Basic Ideas and Exam ples

3

Number of molecules

1000

2000

Velocity/rns-

Figure 1.1 The Maxwell distribution of velocities of molecules in a gas at 273 and 1273 K .

A s

the temperature is increased th e most probable velocity moves t o a higher

value and the distribution widens, reflecting a greater range o molecular velo-

cities.

m otion (translation ) of the a tom s in the gas, the 'translation al energy'.

Remarkably, although the kinetic energy of an individual atom de-

pen ds up on its mass, the prediction is tha t the to tal energy of the gas in

the sample does not.

It

seemed so outrageous w hen first made tha t it had

to be tested, but how? We have calculated the abso lute quantity, E , but

have no way of measuring it directly. But there is a closely related

property that we can measure. This is the heat capacity of the system,

defined as the am ou nt of heat required to raise the tem peratu re of a given

qu an tity of gas (here

1

mole) by 1 K. Different values are ob tain ed if this

measurement is made keeping the volume of the gas constant (with a heat

cap acity defined as C,) o r keeping its pressure co nstan t (C,) since in th e

latter case energy is expended in expanding the gas against external

pressure. H ere we consider jus t w hat is hap pen ing to the energy

of

the gas

itself, and must use the former. Writing the definition in mathematical

form, as a partial differential (a differential with respect to just one

variab le, here T),

(1.2)

, =

g),

lT),

( R

T,

= R

= 12.47

J

K - mol-I

The subscript on the bracket reminds us that we are dealing with a

constant volume system.

This is again rem arkable. It says that for all mo natom ic gases, regard-


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4

Chapter 1

less of their precise chemical natur e o r mass, the m ola r heat capacity is

the same, a n d independen t of temp erature. Experiment show s this to be

correct. For example, He, Ne, Ar and Kr were early shown

to

have

precisely this value over the temperature range 173-873

K,

the range

then investigated.

It is now wo rth exam ining in more detail where the result tha t the m ean

energy of a m ona tom ic gas is indepen den t of its natur e com es from. T o

ob tain this average ove r the whole range of velocities we m ust mu ltiply

the kinetic energy of a n a to m a t a given velocity by the prob ability tha t it

has this velocity,

normalised t o the

tion (dN /N) is the

where

and integrate over the whole velocity distribution

total number of atoms present. This probability func-

Maxwell dis tribu tion of velocities.

Th us the expression for E , clearly and understanda bly, contains the mass,

m,

and the velocity, u. Yet due to its mathematical form and since the

integral is real (an d is evaluated between these up per an d low er limits) its

value ($kT for m otion in three dimensions, Eq uatio n 1.1) does no t. The

expression m ay look formidable bu t th e integral has a stand ard form (it is

a Gaussian function) and its eva luation is straightforward; see Appendix

1.1. It follows tha t the sam e result is obta ined for any form of energy (no t

necessarily translational) that can be expressed in the sa me m athem atical

form,

+ab2,

where ‘a’ is a co ns tan t an d

‘b’

is a variable tha t can take any

value within a Maxwellian distribution. Such a term is known as a

‘squared term’.

F ro m experience, gases are hom ogene ous and possess the sam e prop-

erties in all three directions in space; for example, the pressure is the sam e

in all directions. The mo tion of the ga s ato m s in the three pe rpendicular

Cartesian directions is independent and we say that they have three

‘tran slatio na l degrees of freedom ’. Resolving the ve locity in to these direc-

tions and using P ytha gora s gives, with obvious no tation,

with an analogous result for their means. In the gas the mean square

velocities in the three directions are equal. The form of the Maxwell


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5

distribution we have used is tha t for m otion in three dimensions an d the

average energy associated with each translational degree of freedom is

consequently one-third of the value obtained . It then follows that for each

degree of freedom

whose energy can be expressed as a squared term

we

shou ld expect an average energy of

i k T

per atom . This is an im portant

result of classical physics and is the q uantita tive statem ent of 'the Prin -

ciple of the Eq uip artitio n of Energy'.

W e stress that it h as resulted from the Maxwell distribution in which

there is no restriction of the translational energy that an ato m (o r mol-

ecule) can possess. F ro m everyday experience this seems em inently rea-

sonable. We can indeed make a car travel at a continuous range of

velocities w itho ut restriction (an d luckily perso nal cho ice of how ha rd we

press the accelerator ra the r th an collisions m ake a w hole range possible if

we consider a large number of cars ). But is this true of molecules that

migh t possess othe r sources of energy besides translation ? We sh ould no t

assume

so,

but again put it to experimental test. We shall find later that

we have to re-examine the case of translation al energy too.

1.3

HEAT CAPACITIES

OF

DIATOMIC MOLECULES

Th e heat capacity, C,, of a samp le is directly related to its energy, an d can

be measured. We expect gaseou s diatom ic molecules, like atom s,

to

move

freely in independent directions in space

so

that translational energy

should confer upo n th e sample a heat capacity

of R

= 12.47

J

mol-' K- .

If this was the on ly source of energy tha t m olecules possess then the hea t

capacity shou ld have this value, and be independe nt of tempe rature. This

turns ou t to be w rong on both counts. Fo r example, the m easured heat

capacities (in

J

mol-1

K-') of

dihydrogen and dichlorine at various

tempe ratures are given in Ta ble 1.1.

All these values are substantially greater than expected from transla-

tional motion. Through the direct relationship between C, and

E

this

implies tha t there must be additional con tributions to the energy of the

sample. W e no te also tha t for each gas th e value increases with tem pera-

ture, with a tendency for it to becom e consta nt a t high tem peratures for

Table

1.1

He at capacities ( J mol-'

K - ' )

of dihydr ogen and dichlorines at

diflerent tem peratures

T ( K )

Molecule

298 400 600 800 1000

20.52

20.8

7 21.01 2 1.30

21.89

c 2 25.53 26.49

28.29

28.89

29.10

H2


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6 Chapter

1

dichlorine, an d t ha t over the whole range of temp erature s in the table the

heat capac ity of dichlorine exceeds tha t of dihyd rogen.

So what forms of energy can a diatom ic molecule have that an ato m

can not? Th e obv ious physical difference is th at in the m olecule the ce ntre

of mass is n o longer centred on the atom s. This implies that if there a re

internal m otion s in the m olecule a s it translates through space these have

associated energies. The first, and most obvious, possibility is that the

molecule might rotate. A sample containing rotating molecules might

therefore possess both translational a n d rotatio nal energy, and we need

to assess the latter. The simplest, and quite goo d, model for molecular

rotation is to trea t the diatom ic molecule as a rigid roto r (Figure 1.2)with

the atom s as point masses

( m ,

and

m,)

sepa rated from the centre of mass

of the molecule by distances r l and r 2 . Classical physics shows the

rotation al energy to be

02,

where

I

is its mom ent of inertia an d ci the

ang ular velocity (measu red in rad

s-').

We imm ediately recognise this as a

'squared term'.

Rotation might occur about any of three independent axes which in

general might have different m om ents of inertia, althou gh for a d iatom ic

molecule two are equal. Taking the bon d as on e axis

( z ) ,

hese are those

about axes perpendicular to it through the centre of mass and their

m om ents of inertia a re defined by

(9

(ii) (iii)

Figure

1.2 Rota tions o j a diatomic molecule. T he atoms are treated as point masses, m , and

m2,

with their centres lying along the axis

of

the molecule with th e distances

r l

and r2 measured hetween these centres and the centre of mass ( C M ) of the

molecule. The molecule can r otate about three axe s, in the plane of the paper (i),

about the bond axis

(ii)

and out o the plane

(iii).

The moments o inertia for

rotations

(i)

and

(iii)

are non-zero and equal but the moment of inertia o r rotation

(ii)

is zero since there is no perpendicular distance between the point masses and

the C M along the bond axis. This rotation therefore does not contribute to the

total rotational energy

o the molecule.


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Some Basic Ideas and

Examples

7

We note th at the distances are measured in the direction p erpendicular to

the axes of rota tion , here along th e z-axis. This implies that the mo m en t

of

inertia for rotation ab ou t the z-axis is zero because the point masses and

the centre of mass all lie on a straigh t line, an d n o perpendicular distance

in the

x

or

y

directions separates them. We conclude th at only two of the

three rotational degrees of freedom contribute to the energy of the

molecule, both throu gh squared terms in the an gular velocity. Using the

Equipartition Principle we predict that their contribution to the energy

will be

2 x i R T J

mol--'. Th is implies th at , toge ther with the translation al

con tribution , the tota l energy of the molecule sho uld be

:RT J

mol-land

C,

should be

R

J mol-' K-l. It should no t vary as the tem perature is

changed.

This has the value

20.78 J

mol-I

K-',

which, interestingly and signifi-

cantly (see later), is very close to the v alue ob served for dihyd rogen a t 350

K, but Table 1.1 shows

C,

to increase with temperature. However, for

dichlorine it is still much too low at this temperature compared with

experiment. Once again we conclude that the actual energy is greater

than we thought, and that the molecule must have another form of

interna l mo tion associated w ith it. Th is is vibratio n.

In a vibration the atoms continuously move in and out about their

average positions (Figure

1.3).

As they move outwards the bond is

stretched, as would be a spring, an d this gen erates a restoring force, which

i Equilibrium

position

+ f - -

Figure 1.3 Vibr ation of a diatomic molecule. T he diagram shows at th e top the (point mass)

atom s at their distance o closest approach when the y start to move apart again,

in the centre at their average positions, and at the bo ttom when the bond

is

fullest

stretched and the elasti city

o

the bond brings the a tom s back towards each other

once more. A t the two extrem e positions the a toms are momentarily stationary

and the molecule possesses the potential energy obtained from stretching

or

compressing the bond o nly, but they then start to move, transforming potential

energy into kinetic energy, a process complete just

as

the atoms pass through

their equilibrium positions.

I f

the bond stretching obeys Hooke's Law (the

restoring for ce generated by moving aw ay fr om the equilibrium position is

proportional to the distance moved) then Simple Harmonic M otio n results.


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8

Chapter 1

if Hooke's Law is obeyed is prop ortion al t o the displacement from the

equilibrium positions, an d the ato m s return th roug h these positions. In

this model (also quite good ) the m olecule behaves as a simple harm onic

oscillator with co ntinually interch ang ing kinetic

(KE,

from the m otio n of

the atoms) and potential (PE, from stretching the bond) energies. The

total energy is the sum of the two, a nd is conserved in a n isolated gas

molecule.

Evib= ( K E + PQ

At a ny instan t the k inetic energy is given classically by m2 where ,u is the

'reduced mass' of the molecule (defined as

m, m2/ (m ,

+

m2))

nd

v

is the

instan taneo us velocity of the a tom s, whilst the potential energy is i k x 2 ,

where

k

is the bo nd force con stant (Hooke's Law co nstan t) an d x is the

instantaneous displacement from the average position of each atom. A

diatomic molecule can only vibrate in one way, in the direction of the

bon d, but because

of

having to sum the contributions from bo th forms of

energy this one degree

of

vibrational freedom contributes two squared

terms to the total energy, throu gh the Equ ipartition Principle, 2 x

RT J

m ol? On ce again we have assumed that, in using this Principle, there

are no limitations on (now) the vibrational energy that a molecule can

possess.

Th e total energy of the molecule is, therefore, pred icted t o be the su m of

the translational (;RT), rotational (RT)and vibrational (RT)contribu-

tions, giving ZRT

J

mol- ' and

C,

=

ZR

=

29.1

J

mol-'

K- ,

greater than

before but still independent of temperature. This is precisely the value

obtained experimentally for dichlorine at 1000 K but it is much higher

tha n tha t of dihydroge n at the same temperature. The heat capacities of

bo th are still predicted, wrongly, to be indep end ent of temp erature .

It is now instructive to plot C, against T for a diatomic molecule

(shown diagrammatically in F igure 1.4).The value jum ps discontinuously

between the three calculated values, corresponding to translation alone,

translation plus rotation an d finally translation plus rotation plus vibra-

tion, over small temp erature ranges (near the characteristic tem peratures

for rotation an d vibration, Or and

Ov ib

Section

2.5.1).

These temp eratures

depend on the precise gas studied, and the changes occur at higher

tem peratu res for molecules consisting of light ato m s tha n for those t ha t

contain heavy ones. Only a t the highest tempe ratures are the values those

predicted by E quipartition. But the contribution from translation alone

is evident at tem peratu res close to abso lute zero, bu t n ot extremely close

to it when this contribution falls to zero. In this plot the translational

contribution is easily recognised through its unique value but which

of


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Som e Basic Ideas and Examp les

9

Figure 1.4

rotation

3 5

2.5

1.5

Cvf R

Vibration

otation

Translation

ii

293

0

ot

TIK

Schematic diagram of how

C,

for a diatomic molecule varies with temperature. I t

rises sharply fr om near 0 K and soon reaches 3R/2, expected for translational

mo tion, where it remains until a higher temperature (near O r , , is reached when

the rotational degrees offre edo m contribute another

R

to th e overall value. T his

happens well below room temperature (293 K ). fo r all diatomic molecules. A t still

higher temperatures the vibrational motion eventually contributes another

R,

again near a rather well-defined temperature (

O v i b .

Th e actual values

of

these

temperatures vary with t he precise molecule concerned, and are lower fo r heavy

molecules (e.g. Cl ,) than light ones

(e.g. H 2 ) .

Th e beginnings of the vibrational

contribution occur below room temperature for C1, but the u ll contribution is not

apparent until well above it.

or vibration con tributes at the lower temp erature is obtainable

only through further experiment or theory; the rotational contribution

appea rs at the lower temperature.

W e conclude tha t molecules exhibit very different behaviour from th at

we predicted using classical theory an d we m ust exam ine where we might

have gon e wrong. All the basic equations for rotation al an d v ibrational

energy are well established in classical physics and ar e no t as sum ptions .

O n e possibility might be th at the rotationa l an d vibrational energies are

correctly given classically but that they do not have Maxwellian dis-

tributions. But also we have made what in the classical world seems a

wholly unexceptional assu mp tion,

i.e.

that there are no restrictions a s to

the energies a molecule m ight possess in its different degrees of freedom.

Since the predictions d o not conform t o the experimental observations

this might be wrong.

W e speculate that rather th an being able to possess

any

values of their rotatio na l an d vib rationa l energies, the molecule may

be able to possess only specijic values of them . Th is is confirmed directly

by spectroscopy, see below. W e describe the energies as ‘quantised’. This

is the basic realisation from which mu ch

of

physical chemistry flows.

Translational energy seems no t to be quantised bu t ac tually is; experi-

me nts have to be performed a t very close to 0

K

to observe this. It is why


22/140

10 Chapter

1

C, in Figure

1.4

goes to zero a t this tem peratu re. The fact th at different

diatom ic molecules possess the tw o further forms of energy ab ove cha rac-

teristic temp erature s th at differ from o ne molecule to the next is an aspect

of energy-quantised systems that we shall have to understand. But all

systems behave classically at high enough temperatures, which may,

however, be below room temp erature. F o r example, all diatomics (save

H2)exhibit their full rotational contribution below room temperature.

But only the heaviest molecules exhibit the full vibratio nal co ntributio n

below very high temp erature s.

T ha t the lim iting classical beh aviou r is often observed in real systems,

especially for po lyatom ic molecules, is ultimately w hy we are norm ally

unaw are of the quantised natu re of the world th at su rroun ds us. But the

world is quantised in energy and we need t o und erstand an d exploit the

properties

of

matter that this implies. Much of the new technology in

everyday use depends o n it.

It

is fascinating and significant that this conclusion of paramount

importance was indicated as a possibility through the interpretation of

classical thermod ynam ic measurements, emphasising tha t a conne ction

exists between the therm ody nam ics of systems an d the prope rties of the

individual atom s and molecules tha t com prise them.

1.4

SPECTROSCOPY

AND QUANTISATION

Q ua nt isa tio n of energy show s itself very directly in the optica l sp ec tra of

atoms and molecules and initially we consider the electronic spectra of

atom s. Wh en a sam ple of ato m s is excited in a flame, for example, it emits

radiation to yield a ‘line spectrum’ (Figu re 1.5). This is in fact a series of

images of the exit slit in a spectrometer corresponding to a series of

different discrete frequencies of light emitted by the atoms. If the atom

behaved classically this would n ot be

so

since the electrostatic a ttrac tion

between the electron an d nucleus w ould accelerate one tow ards the other,

an d acc ording to classical physical laws the ato m would em it light over a

continuous frequency range until the electron was annihilated on en-

cou ntering th e nucleus. Th e explan ation is tha t the energies possible for

the electrons in a n a tom are themselves quantised, and the frequencies in

the emission spe ctrum co rrespond to the electron jump ing between levels

of different energy, acco rding to the B ohr co ndition ,

where h is Planck’s con stant,

v

the frequency of the light and

E ,

and

E ,

are the energies of tw o of the levels (Fig ure

1.6).


23/140


11

Figure 1.5

Figure

1.6

120

100

k/nm

Ultraviolet region of the emission spectrum

of

hydrogen atoms (the Lyman

series). Atoms in the sample have been excited by an electric discharge to a

number of higher atomic energy levels and they emit light at digerent discrete

,frequencies as the electrons return t o the lowest level (principle qua ntum number,

n

=

. Th is displays the quantised nature o the atomic energy levels directly.

Other series

of

lines are observed in the visible and infrared regions

of

the

electromagnetic spectrum, corresponding to electron jumps to diferent lower

quantised states. The positions

of

the lines allow the frequ encie s

of

the emitted

light t o be measured.

The origin of a line in a spectrum o frequency

v.

In a H atom at thermal

equilibrium with its surroundings the electron is in the lower energy level and it

can absorb the specific amount of energy hv to ju m p t o a higher energy level. I f

however, the atom has been excited

so

as to put the electron in the upper energy

level then it can emit the sume energy and return t o the lower level. I n a real atom

there are many pairs of energy levels between which spectroscopic transitions can

occur.

An early trium ph of the Schrod inger eq ua tion was tha t it rationalised

why the levels are quantised and allowed the energies to be calculated,

with the difference frequencies agreeing with the observed ones.

Asso-

ciated with each level is a mathematical function, the wave function,

known as an orbital .

The C, measurements discussed above gave no indication tha t m on-

atom ic gases could possess electronic energy besides translationa l energy.

This must m ean that

ouer

the

temperature range studied

the atoms do not

hav e any. But we have seen in Fig ure 1.4 th at d ifferent sources of energy

m ake their contributions a t different tempe ratures and at a high eno ugh

temp erature there would indeed be an electronic con tribution to the

C ,

of atom s. We shall see later th at the crucial factor is the energy sepa ration

between the qu antised energy levels com pared with th e ‘thermal energy’

(given by

k T )

in the system, and the lower atom ic orbitals are separated

from each other by large energy gaps. For some atoms, such as the

halogens (see below) an electronic contribu tion is evident even a t q uite

low temperatures, bu t this is rathe r unusual.

In molecules, too, discrete energy levels and molecular orbitals exist

with differing electronic energies. But associated with each and every


24/140

12 Chapter

1

electronic level there is a set of vibrational and rotational ones (Figure

1.7).

Th roug h the Bo hr condition spectroscopy allows us to measure the

energy gaps directly, whilst q ua ntu m mechanics also allows us to calcu-

late the energy levels, an d the ga ps between them. T his confirms tha t the

energies molecules possess are quan tised. Molecular sp ectra are norm ally

observed in a bso rption , with the m olecules absorbing certain frequencies

from white light flooding through a sample of them. Whereas emission

spec tra arise w hen electrons fall from higher energy levels into w hich the

substance has been excited by heat or electricity, absorption spectra

dep end on species in their lower energy levels jum ping to h igher ones.

They therefore give inform ation o n th e lowest energy levels of the mo l-

ecules. Experim entally a huge range of trans ition frequencies is invo lved,

varying from th e m icrowave (far infrared) to the u ltraviolet regions of the

electromagnetic spectrum. The former occur between energy levels that

are closest in energy, those d ue t o rotatio n. At higher frequencies, in the

infrared, the light has sufficient energy to cause jumps between vibra-

tional levels, bu t these all have m ore closely spaced associated rota tion al

ones, an d under certain selection rules changes occur to both the vibra-

tional an d rotatio nal energies simultaneously. Th e spectra are know n as

‘vibration-rotation’ ones (an example is given in F igure

2.6,

Chapter 2).

Finally, at mu ch high er frequencies, in the ultraviolet, electrons can ju m p

between the vibrational and rotational levels of different electronic en-

ergy states and the sp ectra reflect sim ultaneous changes in all three types

ot

energy.

Through Equation (1.8) there is a direct relationship between fre-

quency an d energy difference an d we conclude tha t in terms of the gaps

between the energy levels

AE(e1ectronic)>> AE(vibration) > AE(rotation)(and

>>

AE(trans1ation)) (1.9)

This confirms o u r conclusions from

C,

measurements (Figure

1.4).

At

0 K

Cv is zero, but as the temperature is increased translational motion

rapidly makes the contribution expected from classical physics. At a

higher temperature the effects of rotation become apparent, and at a

higher one still, vibration (and at very high temperatures electronic

energy contributions might appear if dissociation does not take place

first).

M oti on s corresponding to th e lower energy gaps make their contribu-

tions at lower temperatures than those with higher energy gaps.

It is important to distinguish between the absolute values of the

various types of energy and the separations between the energy levels.

Th us the g aps between tran slationa l levels are miniscule but even at very

low temperature there is a contribution of ( )RT mol-‘ to the energy.


25/140

Some

Basic

Ideas and Examples

13

Energy

I

Electronic

Vibrational Rotational

Figure

1.7

Schematic diagram o the energy levels o a diatomic molecule showing on ly the

two lowest electronic levels, which are widely separated in energy. Associated

with each is a stack o more closely spaced vibrational levels, which in turn

embrace a series o rotational levels. Under the Simple Harmonic oscillator

app rox imation, the vibrational levels of each electronic level are equally spaced,

but the rotational levels predicted by the rigid rotor model are not. T his diagram

gives an impression

of

the relative sizes

o

the electronic, vibrational and rota-

tional energies but o nly a very f e w o f t h e lower energy levels can be shown here

without loss o clarity . Rea l molecules have fa r more levels. Spectroscopic

transitions can be excited between the rotational levels alone, between the

rotational levels in diferent vibrational states and between the sub-levels

o

the

electronic energy sta tes. T he three ty pes occur with very diflerent energies and

are observed in quite di fer ent regions of the electromagnetic spectrum.

Th e gaps between electronic energy levels are en orm ous in co mp arison,

yet electronic energy makes no contribution to the to tal energy of most

systems a t room temperature.

A

com m on er ror is to believe that since the

energies of molecular electronic levels may be high then the electronic

energy of the system must be high. If the molecule existed in one of the

higher levels the energy would indeed be high. But it does not at low

temperatures, where the molecule is in its lowest electronic level. F o r a

gas at room tempe rature the translational energy is greatest in magn itude

followed by the rotation al energy and then the v ibrational one. The o rder

is exactly reversed from th at

of

the energy gaps:

E(translationa1)

>

E(rotationa1)

>

E(vibrationa1)

>>

E(e1ectronic))

(1.10)


26/140

14

Chapter 1

1.5 SUMMARY

Atom s an d m olecules may possess several different con tribution s t o their

total energy b ut each one ca n have only certain discrete quantised values.

Th e energies associated w ith different m ode s of mo tion differ in m agn i-

tude, with the g ap s between the energy levels for translation being sm aller

than those for rotation that, in turn, are smaller than those for vibration.

The gaps between the electronic energy levels of molecules normally

vastly exceed all of these. This causes translational motion to occur at

lower tem peratures than rotation al mo tion, which occurs a t lower tem-

peratures t ha n vibration (and much lower than electronic excitation). Yet

we remain u naw are of quantisation in every day life, an d we have som e

indication already that this is inherently because systems behave classi-

cally at high en ough tempe ratures. W e need to sha rpen this concept and

decide w hat a ‘high enough ’ tem pera ture is, an d then we sho uld be able to

predict the behaviour

of

systems from a knowledge of the quantised

energy levels of the at om s an d m olecules from w hich they are m ade .

1.6

FURTHER

IMPLICATIONS FROM SPECTROSCOPY

We are so familiar today with spectra that we tend to miss a very

remarkable fact about them. An atomic spectroscopic transition in ab-

sorptio n, for example, occurs when an ato m in a specific energy level

accepts energy from the radiation an d jum ps to ano ther specific level, in

accordance with the Bohr condition and under various selection rules

th at limit the transitions th at a re possible. These have been discovered by

experiment, and can be rationalised using quantum mechanics.

So

the

spec trum of a single at om und ergo ing a single transition is a single line at

one specific frequency. But w hen we talk abo ut the spec trum of a n ato m

(a loose term ) we imm ediately think of a w hole family of transition s a t

different frequencies (Figu re

1.5).

Since the electron in on e ato m ca n only

m ake one jum p between energy levels a t a time it follows that wha t we see

is th e result of a large number of individual atom s simultaneously absorb -

ing energy and jump ing t o a whole range of possible q ua ntu m states. Th at

is, instead of seeing the sp ectrum of a single ato m , we a re observing the

spectra of a very large number of individual atoms simultaneously.

We

say that spectroscopy is an

ensemble phenomenon,

meaning precisely tha t

w hat we observe is the result of wh at

is

hap pen ing in the large collection

of atoms.

But w hat w ould h ap pen if we were clever eno ugh t o observe a single

atom over

a

long period of time, rath er tha n instantaneously? Following

the initial abs orptio n of energy from the light beam , the a tom enters a


27/140

Some Basic Ideas and Exam ples

15

higher energy level. Let us now postulate that an efficient mechanism

exists (it does ) for returnin g it t o the lowest level, where it m ight a bs or b a

different frequency from the incident radiation and attain a second,

different, higher level. It w ould then retu rn an d the process be repeated

over a whole cycle of possible transitions covering the total range of

frequencies. We conclud e th at over infinite time the a to m w ould perform

all the possible transitions allowed to it and the spectrum of the single

atom would be identical in appearance to the ensemble one. This has

been pu t to d irect experimental test in recent years, alth ou gh in mo lecular

rather than atomic spectroscopy, and found to be correct. It opens the

possibility of calculating ensem ble behavio ur of a collection of molecules

from the b ehaviou r over time of a single one, a concept close to the basis

of using ensembles in statistical therm ody nam ics (see later).

A noth er unexpected asp ect of spectroscopy lies in the Bo hr cond ition.

We tend to think of atoms or molecules absorbing energy from an

incident light beam and jumping to higher energy states, but Equation

(1.8) does n ot dictate a direction for the energy change to occur. T h a t is,

whilst an atom in a lower energy state might jump to a higher energy

state, one in th at s tate might em it energy unde r the influence of the light

beam , and fall back t o the lower state. In this case the beam would exit

more intense than it arrived. These processes are known as stimulated

absorption and stimulated emission respectively. Einstein w as th e first t o

consider this a nd showed by a simple kinetic argum ent (it is now mo re

satisfyingly don e using qu an tum mechanics) tha t the abso lute probab ility

of an upward or a downward transition caused by light

of

the correct

frequency is exactly the sam e. It follows that if there a re m olecules in b oth

energy levels then the intensity of a spectroscopic line depends on the

difference between the nu mb er of a tom s th at ab so rb energy from th e light

beam an d those that emit energy to it, a n d therefore on the

diflerence

in

the populations of the two energy levels. This is further considered in

Section

4.4.

So not only can spectroscopy measure the energy gaps in

ato m s and m olecules, bu t it indicates this difference in popu lations, to o.

Since all atoms and molecules at thermal equilibrium with their sur-

roundings are found experimentally

to

exhibit absorption spectra, we

conclude tha t

at

thermal equilibrium the lower energy states are the mo re

highly populated. Samples in which the a tom s are d eliberately excited to

higher levels, for example, by an electric discharge through them, have

their upper levels overpopulated and exhibit emission spectra. This is

how streets are lighted, using the emission spectrum of sodium. We

should always remember that systems are not necessarily at thermal

equilibrium; indeed, equilibrium can be disturbed or avoided in many

ways

of

increasing technical im portance. F o r example, lasers depen d on


28/140

16 Chapter

1

deliberately prod ucin g o verp opu lations of higher energy states.

1.7

N TURE OF QU NTISED SYSTEMS

Th e quan tised world is a strange a nd un-instinctive one, complicated by

the fact that polyatomic molecules possess millions of discrete energy

levels. But we can understand wh at q uan tisation implies by study ing it a t

its simplest and we initially consider a system that has just two energy

levels available to it. Th a t is, one in w hich the energy can not vary w ithout

restriction, as in the classical physical world, but in which the atoms,

molecules, nuclei or whatever t ha t com prise the system can individually

ad op t on e of jus t tw o possible energy states. Such systems actually exist.

F o r example, the nucleus of the hydrogen atom , the pro ton, is magnetic

and can interact with an applied magnetic field to affect its energy.

Experiment shows that it is a q ua ntu m species whose magnetic mo me nt

can adopt either of two orientations with respect to the field direction,

rather t ha n the one tha t a classical com pass needle would. In o ne orienta-

tion the m agnetic m om ent lies alon g the direction of the applied field, an d

its energy is lowered, whilst in the oth er it o ppo ses it, an d its energy is

increased. These a re simple experimental facts an d they imply th at appli-

catio n of the field creates a tw o-level system [F igu re 1.8(i)]. Th is is

exploited in Nuclear M agnetic Resonance (N M R ) spectroscopy, in which

transitions are excited between the two levels. A

different example is

found in the halogen atoms that behave as though they were two-level

systems a t low tempe rature.

Let us be clear what is implied by the existence of the two levels. We

define the energy of the low er to be

0,

an d that of the upper on e

E

If we

have one particle (an atom ,

a

nucleus o r w hatever) in its lowest level its

energy is 0, whereas if, som ehow , we pu t it in to the upp er s tate its energy

is E It is crucial to o u r understanding of qu antised systems that there is no

other possibility. Th e particle ca nno t, for example, have a n energy

of c/2

or 1.28. This seem s at od ds w ith everyday life in which, up to som e limit,

systems seem to be ab le to possess any energy. F o r example, we can h eat a

kettle to any temperature below the boiling point of water. We need

somehow to reconcile this difference with the classical world since we

know that a t atomic o r molecular level

all

energies are quan tised.

Th e secret once mo re lies in the fact th at in the exp eriments we usually

perform we do not study individual particles but rather collections of

them in which they may be dis tributed between the two energy states. W e

sta rt by simply add ing a seco nd [Figure 1.8(ii)]. N ow bo th m ay be in the

sam e energy level, to give a tot al energy of 0 o r 2c or on e may be in one


29/140


17

Magnetic Field

- I

0

(ii)

Figure 1.8

(i)

A two-level system results when hydrogen nuclei are placed inside a magnetic

field. Some protons align with their magnetic moments along the applied j e l d ,

and some against it, resulting in two difSerent energy states. At thermal equilib-

rium there are more nuclei in the lower energy state than in the upper one.

Nuclear Magnetic Resonance Spectroscopy consists in causing a transition

between the two.

(ii)

If just two particles enter

a

two-level system their total

energies can be

0 , 2 ~r E,

but i ft h e y enter the levels with equal probability then

the latter can be obtained in two way s, unlike 0 and 2~ which can be obtained in

jus t one.

level and the other in the other, giving a total of

E .

Now consider the

average energy of the two. F o r this latter case this is &/2,

so

tha t we now

have an energy tha t is not one of the quan tised values. Th e total energy is

simply the n um ber

of

particles times the average value

(2

x

~ / 2 ) , nd in

this case has a value equal to th at

of

one

of

the qua ntised levels. But this is

rarely true as we increase the num ber

of

particles in the system. Co nside r

one in which there are

( m +

n) particles divided between th e tw o energy

states, with

m

in the lower energy one. N ow the total energy is

nE,

which

can take a range

of

values determ ined by n, an d the average energy

(1.11)

which is determined by the values of m and n. W hereas the energies of the

individual particles have discrete, quantised, magnitudes, the average

energy, and the total energy, have no such restrictions and can take a

wide range

of

values. This lies at the heart of why systems containing


30/140

18 Chapter 1

species with q uantised energy levels nevertheless display classical behav-

iour und er m ost c onditions. These energies clearly depend on how the

particles are distributed over the two energy states, tha t is on the num bers

m

and

n.

This simple example leads to a further important insight. If we can

distinguish between the particles,

i.e.

know which is which, then the

situations with

0

o r 2~ in total energy can each be obtained in just one

way. But an energy of E is obta ined if either is in the low er level an d th e

oth er in the upp er o ne, in tw o w ays. If b oth levels are equ ally likely to be

populated this energy is twice as likely to occur as either of the others.

Generalising, this implies that som e population distributions a nd some

energy values ar e m ore likely to occur tha n others. This is

a

conclusion of

mo men tous im portance. However, we stress tha t it depend s on the likeli-

hood

of

pop ulating each level being inherently equal.

We shall now consider systems containing

a

large number of, e.g.

molecules, which may exist in any of a large number of energy levels,

before returnin g t o the two-level system.

1.7.1

Boltzmann Distribution

Within chemistry we habitually deal with systems that contain a very

large numb er ( N )of molecules an d we consider how these are distributed

between their numerous energy levels when the systems are in thermal

equilibrium with their surroundings at temperature T. For example,

1

mole of gas at 1 bar pressure contains N ,

(6.022

x molecules. W e

need the q ua nt um analo gue of the M axwell distribution of energies. It is

given by the Boltzmann distribution, which we shall state and use here

before deriving it in Chapter

2.

It is a statistical law that applies to a

constant number

of

independent non-interacting molecules in

a

fixed

volume, an d is subject to the to tal energy

of

the system being consta nt

(the system is isolated), an d the several ways

of

obta ining this energy by

distributing the molecules between the quantised levels being equally

likely. That is, it does not matter which particular molecules are in

specific energy states provided tha t the total energy is con stant. The

distributio n is

(1.12)

where ni s the number of molecules in a level of energy ii and

g i

s the

degeneracy

of

that level (the number

of

states

of

equal energy, for


31/140

Some Basic I d e m and Example s

19

example, the three p o rbitals

of

a

H

ato m have the same energy and

so for

this

g i

=

3).

The denominator includes summation over all the energy

levels of the molecules. This equation results from statistical theory

applied t o a very large num ber of molecules, und er which condition s on e

particular distribution of the molecules between the quantised levels

becomes

so

mu ch m ore likely than the rest that it alone need be consider-

ed. This is the ultimate extension of ou r conclusion concerning just two

levels.

O n first encounter, the B oltzman n d istribution looks formidable, es-

pecially because it apparently involves summation over the millions of

energy levels present in po lyatom ic molecules. But we now return to the

two-level system to discover the circumstances where this is only an

ap pa ren t difficulty.

1.7.2 Two-level Systems

Consider a two-level system in which there are no particles (atoms,

molecules or wh atever) in the lower level an d n l in the upper on e so that

the total number N

=

(no+ nl). At therm al equilibrium the Boltzmann

distribution tells us that these are related th roug h

Rea rrangem ent yields

(1.13)

(1.14)

In the simplest case the degeneracy of each state is 1, e.g. for protons

inside a magnetic field. H ere the ratio of the po pulatio ns dep end s directly

an d solely on the value

of

the dimensionless expon ent (@T) ha t varies as

the temp erature is change d, being a fixed characteristic of the system.

The denom inator,

kT,

is kno wn as the 'thermal energy'

of

the system. This

energy is always freely available to us in systems a t therm al equ ilibrium

with their su rroundings a nd, indeed, we canno t avoid it without decreas-

ing the temp erature to

0

K.

Th is gives us a simple physical picture. The

therm al energy is wh at a system possesses by virtue

of

the motion

of

the

particles th at com prise it, an d we see th at it is closely related, for exam ple,

to the mean thermal energy due to translation +kT, above. But the

distribution tells us th at in q uantised systems we must c om pare

kT

to

rather than

2

imes it. If

kT

is mu ch less than

E

we do n ot have the energy


32/140

20 Chapter 1

to raise the particle from its low er energy level to the higher on e, but as

the temp erature is increased it becomes possible t o d o

so.

So much for the basic picture; now let us investigate the distribution

semi-quantitatively. At low temperatures

kT


33/140

Some

Busic

Ideas

and

Examples

21

T T T

0)

(ii) (iii)

Figure 1 9 (i) I n a two-level sys tem all the population is in the lower level at absolute zero

but t he number in the upper level

(n , )

grows as the tem perature is increased. I t

does not grow indefinitely, however, but reaches an asymptotic value at high

temperature.

I f

the levels are singly-degenerate, half of the total number

o

molecules is then in each level.

(ii)

Th e total energy

of

the sys tem is the product

o

the population o the upper level times its energy (see te xt ) and

so

it varies with

temperature exactly as does n l . Th e energy also reaches an asym ptotic value at

high temperature. (iii) Variation o C,with temperature is given by the difleren-

tial with respect t o temperature

of

the energy curve,

(ii)

It starts

from

zero , goes

through a ma xim um at the point o inflexion of the energy curve, and returns to

zero a t high temperature.

This is a simple multiple

of

y1

so

that the variation of energy with

temperature has exactly the same form as the variation of n , [Figure

1.9(ii)]. This is astonishing. It show s th at as th e tem peratu re is increased

E

does not increase continually but again tends to a n asym ptote. T o check

this we must again turn to a calculation a nd m easurement

of

C,, which is

obtained over the temperature range simply by differentiating Figure

1.9(ii). [ C v = (aE/aT),]. Th is is shown in F igu re 1.9(iii). It predicts, what

wou ld be very stran ge in classical physics, th at the heat cap acity increases

through a maximum and then falls

to

zero as the temperature is in-

creased, a n d is wh at is observed expe rimentally.

This precise behaviour is unique to the two-level system. But the

argum ents we have used are no t. Th us the B oltzmann distribution in any

system, in which an y particle m ay exist in an y on e

of

a number of discrete

energy levels, always contains exponential terms in which quantised

energies are c om pared with kT, an d it is the values of these exp onentials

th at largely determ ine level pop ulation s a nd all the physical prope rties of

the sample. That is, they all depend upon the ratio ( /kT).his simple

realisation gives prob ably the most im po rtan t insight in to physical chem -

istry.

An exam ple is seen if we extend th e arg um ent used abo ve to calculate

the energy of the two-level system t o on e with m any levels. W e realise tha t

the particles are distribu ted am ong st the levels each

of

which has its own

characteristic qu antised energy, ii nd we must sum the energies

of

them

all. It follows tha t


34/140

22

E = E n i s ,

1

Chapter

I

(1.19)

This seemingly obvious statem ent h as astounding implications when we

realise what we have done. It says that if we know the values of the

qua ntised energies, which we can determ ine from spectroscopy o r calcu-

late using qua ntu m m echanics, then we can calculate a thermody nam ic

property. It establishes a direct relationship between the properties of

individual atoms and molecules and the thermodynamic properties of

samples made u p of large numb ers of them, an d it is one of the funda m en-

tal equa tions of statistical thermody nam ics. W e shall return t o it later.

1.7.3

Two-level Systems

with

Degeneracies Greater than Unity;

Halogen Atoms

At roo m tempe rature the electrons in ato m s are found exclusively in their

lowest orbitals. Th is is because the highe r orbitals a re greatly separa ted in

energy from them

so

that (&/kT)

>

1.Th is is true of the halogens, bu t w ith

these the lowest level is split into tw o by spin-orbit coupling, which

is

smallest in fluorine and largest in iodine (Figure

1.10).

Such coupling

results because motion of the electrically charged electron around the

nucleus in a p -orbital causes a m agnetic field there th at is experienced by

the electron itself. Ho wev er, the electron possesses spin angu lar m om en-

tum tha t causes it to have a q uite separate magnetic mo me nt. As with the

proton in an external magnetic field the quantised electron magnetic

moment can adopt just two orientations inside the field due to orbital

mo tion, an d two energy levels result. Experiment shows tha t the energy

sepa ration between them is very low com pared with the energies sepa rat-

ing the orbitals

so

that the halogens behave as if they were two-level

systems a t n orma l temperatures.

2p3 2

0

g = 4

Figure 1.10 The lowest electronic energy level of the halogen atoms is split into two by

spin-orbit coupling, the interaction between th e magnetic moments due jirs tly

to the orbital motion

of

the electron in its p-orbital and secondly due to its

intrinsic spin. Inpuorine the splitting in energy is of the order

of

k T a t room

temperature and both levels are populated. The next lowest electronic level is

comparaticely very high in energy (energy

>>

k T ) a n d is completely un-

populated at room tem perature. A t thi s temperature the atoms behaue as though

they are two-level system s, but the t wo levels have difer en t degeneracies.


35/140

Some Basic

Ideas and Examples

23

We take F as an example. Its electron configuration is ls22s22p5 o

tha t it has a single unpaired electron in a p-orbital. The states that result

from spin-orbit coupling can be calculated simply using Term Sym bols

(Appendix 1.2) tha t show the gro und state to split into

2P1,2

nd

2 P 3 , 2

components. The subscript indicates the total angular m omentum qu an-

tum num ber of the state, J , and according to H und's rule the state with

the highest J , 4, ies lowest in energy w hen a n electron shell is over half

full. How ever, the Term Sym bol has an oth er crucial piece of inform ation

encoded in it since the degeneracy of a sta te is (2 J

+ l ) ,

an d we have seen

that the degeneracy enters the Boltzmann distribution. For the upper

state g,

=

2 whilst for the lower o ne

go =

4 (Fig ure 1.10)so that

(1.20)

Th e expo nential term changes with tempe rature exactly as before, tend-

ing to unity as

T

-

00.

In conseq uence the asym ptotic value of the ratio is

no longer 1 but 0.5. Th at is, at high tempe ratures one-third of the ato m s

are in the upper state compared with the half obtained when the levels

have equal degeneracies. Had the state with

J

=

been the upper one

then tw o-thirds of the atom s would have been in the upper state at the

higher temperatures. Simply put, a t high temp eratures, a state of degener-

acy g can hold

g

times m ore atom s than one of degeneracy one -t h e states

behave a s thoug h they were buckets. Deg eneracy has a significant effect

on level populations.

Th rou gh the direct relationship between energy an d heat c apacity it is

clear tha t the halogen ato m s have C, values tha t reflect their ab ility to

accept electronic energy within these split ground state levels besides

possessing trans lationa l energy.

It rem ains to pu t in som e values to see how significant this is. No tably,

the exponent always appears as a ratio so, provided that we express

num erator a nd deno mina tor in the sam e units, i t does not ma tter what

the un its are. Spectroscopists measure

E

using experimen tally convenient

reciprocal wavelength units, den oted an d usually quo ted in cm-' (1

cm

=

m). These are directly related to energy th roug h the relations

E

= hv

and c = v / l o r c =

vV,

where

v ,

1 and c are respectively the

frequency, wave length and velocity of the light. It follows th at

E

=

h c b .

But rather than calculating this each time it is convenient to calculate

kT/hc in cm-I an d to use the measurem ent un its. (A discussion o n units is

provided in Appendix 2.1, Ch apter 2). F o r F,

=

401 cm-l, whilst kT/hc

a t 298 K (room temperature)

=

207.2

ern- ,

so

that

E/kT

=

1.935, and

e - ~ / k T= 0.144. T o work ou t the electronic contributio n to the energy

of

1


36/140

24 Chapter

1

mole of F atoms at this temperature we first return to the Boltzmann

distribution to ob tain the num ber of atom s in the upper state:

g

l e

l k T

2

x

0.144

= 0.067

1

N A

g o e WkT + y lecE ikT 4

+

2

x

0.144

(1.21)

where, as usual, we have defined the lower state t o have zero energy. Th e

total electronic energy per mole at this temperature is

E = n , & N , =

0.067

x

401N, cm-' mol-', wh ich is 328

J

mol-l. Th us the

presence of the low-lying sta te increases the tot al energy of the

system from the pure translation value of 12.47 x 298

=

3716 J rno1-I

(recall

C

=

12.47

J

mo1-I K-' for tran slation) by roughly 9 % . This

increases rapidly with tem pera ture.

The electronic heat capacity is given by

C,

= (dE/dT) = &(dn,/dT),

where the latter is obtained by differentiating the previous eq ua tion .

1.7.4

A

Molecular Example:N O

Gas

NO

is the only simple diatomic m olecule th at contains a single unpaired

electron and it is in a

n*

orbital, implying that it possesses one unit of

orbital angular momentum about the bond axis, yielding a magnetic

mo m ent. Magn etic interaction with the electron spin magnetic mo me nt

once mo re results in spin-orbit coupling (Figure 1.11) and the gro und

state is split into two, giving

2111.,2

and 2113,2 tates, with now the former

the lower in energy. In diatomic molecules, as opposed to atoms, the

degeneracies cannot be assessed from thesesymbols, but each is doubly

degenerate

(go

=

g1 =

2). Th e higher sta te lies 121 cm-' abo ve the lower

so

that a t room temperature

e /kT

.58 and

(1.22)

showing that over one-third of the molecules are in it at room temp era-

ture. As before, we could wo rk ou t wh at this implies as a mo lar contribu-

tion to the tota l energy of the system but it is obviously appreciable, an d

so

is the effect on the heat capacity. But, as with all light diatomic

molecules, there is n o co ntribu tion to the heat capacity a t this tempera-

ture from vibratio nal m otion, since the first excited vibration al level is too

far removed in energy from the g roun d state to be occupied.

Clearly, it is simple t o calculate the e lectronic energy an d hea t capacity

of F and

NO.

But our calculations need not be restricted to these

properties. A sample

of NO

a t room tempe rature is found to be magnetic


37/140

+ - -

t P

b Ps

-

- -

(ii)

Figure

1 11

I n

N O

there is an unpaired electron in a n-orbital w ith orbital angular momen-

tum (with quantum number = 1) about the bond a xis; the vector representing

this is therefore drawn along this axis. (i) Relative to this the spin angular

mome ntum vector o the electron s =4 can lie either parallel to it, to give an

overall momentum of ,or antiparallel to it 4, eading to I IS l 2nd 211112 tates.

(ii)

These motions

o

the negatively charged electron produce magnetic mo-

men ts, also along th e axi s but in the opposite direction to the angular momen-

tum vectors, and experiment shows that the magnetic moment due to spin

motion is (almost) equal to that due t o orbital motion. In the for me r state th e

moments add to make the state magnetic but in the latter the moments are

opposed and the state is not magnetic. I n the molecule, as opposed t o the atom ,

each state is doubly degenerate g =

2)

and, with the next lowest molecular

orbital well removed in energy from the lowest, N O behaves as a two-level

system e xac tly as shown in Figure 1.9.

(actually param agnetic). At first sight this seems unsurprising in a mo l-

ecule tha t co ntains an unpaired electron, since electrons are m agnetic, but

we m ust rememb er th at in spin-orbit cou pling we have discovered the

influence of a second m agn etic field within th e m olecule, due to orb ital

m otion. W e have therefore to consider the resultant m agnetic field of the

two ra ther tha n just tha t of electron spin. By qua ntu m laws these can lie

only parallel (the 21-1312 tate) or antiparallel (the 2111j2tate) to each

oth er alon g the molecular axis. In the former case the m agnetic mo me nts

re-enforce each other, and in the latter they are opposed. It was dis-

covered experimentally that the m agnetic mom ent d ue t o the spin m otion

is

almo st exactly equal (to 0.11 ) to tha t due to orbital motion so that the

two cancel in the

2111,2

tate, m akin g it essentially non-m agnetic. Since

this is the lower energy sta te all the molecules w ould be in it a t sufficiently

low temperature and the sample would not be appreciably magnetic.

Th at a roo m tem perature sam ple is magnetic results from therma l popu-

lation of the up per state.

25


38/140

26

Chapter 1

A

measure of the magnetism of a bulk sample is its susceptibility

x,

Greek chi) which is again straightforward to calculate. Calling the mag-

netic m om ent of

a

molecule in the 2113 2tate

p,

it is given by

and n,,z per mo le is obtained from the B oltzmann distribution as before

(but remember

go = g1

here). Th e tem peratu re dep endence of the suscep-

tibility ha s exactly the sa m e m athem atical form as d oes the pop ula tion of

the upper state, an d th e energy of the system. It therefore increases from

(near) zero at

0 K

an d rises to an asym ptotic value. We see how powerful

some rathe r s traightforward ideas are in calculating the physical pro per-

ties of collections of ato m s and molecules.

APPENDIX 1.1

THE

EQUIPARTITION INTEGRAL

From Equations

(1.3)

and

(1.4),

Since the stan dard integral

(1.24)

(1.25)

we find th at th e terms in m cancel and

mu2

= SkT

(1.26)

This result obviously holds for an y ‘squared term’ whose energy distribu -

tion

is

given by the Maxwell equa tion.

APPENDIX 1.2 TERM SYMBO LS

A Term Symbol provides a straightforward means for assessing what

states of an ato m exist as a result of spin-orbit coupling from kn owledg e

of the electronic structure

of

the atom . In general an atom possesses many

electrons and coupling occurs between their spin and orbital motions

according to definite rules. For light atoms (those with low atomic

numbers) Russell-Saunders coupling decrees that the spin and orbital

angular m om enta sum according t o the rules:

s csi

I

(1.27)


39/140

Some

Basic Ideas

and

Examples

27

L = p i

I

(1.28)

where

S

is the vector sum of all the individual spin vectors

of

the

electrons,

si

nd likewise for

L ;

the qua ntum unit of angular mom entum

(h/2n) is omitted by con vention. Th e total angular mo me ntum resulting

from coupling between the magnetic moments due to these resultant

m om enta is then given by

J = L + S

(1.29)

A further convention is to write all these quantities in terms of the

qua ntum numbers (scalars) rather than actual values

of

the angular

mom enta, despite having to remember tha t vector addition is involved.

Th e result is expressed in the T erm Sym bol

2 s + lLJ (1.30)

Orbital angular momentum quantum numbers of individual electrons

are given letter symbo ls. Th us

s,

p, d and refer to values of

0,

1 , 2 an d

3

respectively. W hen the vector ad dition has tak en place capital

S,

P,

D

and

F symbols are used for the correspo nding tota l

L

values.

In filled electron shells individual si values cancel, as do the corre-

sponding

i

ones, an d we need consider only the u npa ired electrons. In the

F atom , with one unp aired electron in a 2p-orbital,

S = s i =

and L=

i =

1, which is a

P

state. By the general laws of qua ntu m mechanics the

two vec tors can lie only parallel o r antipara llel to each other, yielding jus t

two possible values of

J ,

4and

.

Their Term Sym bols are 2P3,2nd 2Pli2

respectively, since (2 s + 1)= 2. They are described as ‘doublet P three

halves’ an d ‘doublet P half’ states. Their energy separation depends on

the spin-orbit cou pling con stant, which differs between different halogen

atoms . Fr om general qua ntu m m echanical principles, the degeneracy of

each state is given by ( 2 J + 1).

W ith diatom ic molecules a sim ilar con ven tion is used, except that the

total angular momenta are summarised in Greek alphabet symbols,

C

(sigma), Il (pi), an d

A

(delta) corresponding to S, P and D in atoms. The

basic symbo l becomes

2c

‘A where s analogous to

S

and A (lambda)

to

L,

an d spin-orbit coupling ma y again occur between the two m om en-

ta . Fo r NO with its single unp aired e lectron in a

n*

orbital (A = 1, a ll

state) the resultant states are, consequently, and

2113/2.

ut these

both have degeneracies

of

2, no t wh at we would expect for atom s. They

ar e referred t o a s ‘doublet Pi half’ an d ‘doublet Pi three halves’ states.


40/140

28 Chapter 1


41/140


29


42/140

CHAPTER 2

Partition Functions

Chapter 1established that atom s a nd molecules possess quantised energy

levels and that the energy gaps between them can be measured directly

using optical spectroscopy . If the energy of the lowest state is know n th en

this allows absolute values for the energies of the systems to be ascer-

tained. In all cases but on e, the energy du e to v ibration in a m olecule (see

below), the lowest energy is defined to be ze ro. In addition , assuming the

Boltzmann distribution for the populations of the energy levels in systems

a t thermal equilibrium with their surrounding s allowed us to calculate

the physical properties of some two-level systems. We could take the

agreem ent between results

so

calculated a n d experiment as evidence that

the distribu tion is correc t. But th e theoretical d erivation

of

the d istribu-

tion gives insight into the conditions under which it is valid, and we

return to this below. Meanwhile we continue to demonstrate that the

simple ap pro ach we have used w ith two-level systems can be generalised

to ones con taining an y num bers of levels.

2.1 MOLECULAR PARTITION FUNCTION

Th e energy

of

any quantised system can be ob tained, as stated above, by

summ ing over the popu lations of the individual s tates multiplied

by

the

energies of those states:

This is, though, an impractical formula for estimating E for systems,

including molecules, in which the number

of

energy levels may be very

large indeed. First, it involves a su m m ation over term s in each of these

and, second, we appear to need to know the energies of all of states.

How ever, we have alread y seen this is no t necessarily the case. Since the

populations (n ,)depend on e-&iikT, ny terms in the

expansion of

the

sum

for which ii>> kT have near-zero values of

ni

can be neglected. This

30


43/140

Partition Functions

31

limits the nu mb er tha t needs be considered and has far-reaching conse-

quences.

A common approximation is to assume that the various modes of

energy that a molecule m ay possess are indepe ndent,

so

th at for each level

‘total = Eelectronic ‘vibrational + ‘rotational + ‘translational

(2.2)

This is the sam e assum ption m ade in interpreting molecular spectra. It is

a very good approximation but it is one and there are many cases in

which it can be seen to be (slightly) inaccurate.

A

simple exam ple is tha t a

change in the v ibrational energy of a n an harm onic oscillator (e.g. a real

diatom ic molecule) involves a ju m p to a h igher vibratio nal level in which

the ave rage internuclear distance differs from th at in the lowest one, and

therefore causes a change in the moment of inertia and the rotational

energy of the m olecule, too. H ow ever, the app roxim ation is sufficiently

good for us to be able to understand the properties of molecules and to

calculate their thermodynamic properties, often within measurement

accuracy. If we require their exact values we have no alternative to

me asuring the energy levels experimentally an d e ntering them into the

un-approx imated energy equation, without assuming tha t the individual

contributions are indepe ndent.

Summ ing over the know n energy levels is the mo st convenient way of

obtainin

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